Study of singularities in nonsmooth dynamical systems via singular perturbation

Jaume Llibre, Paulo R. Da Silva, Marco A. Teixeira

Research output: Contribution to journalArticleResearchpeer-review

35 Citations (Scopus)

Abstract

In this article we describe some qualitative and geometric aspects of nonsmooth dynamical systems theory around typical singularities. We also establish an interaction between nonsmooth systems and geometric singular perturbation theory. Such systems are represented by discontinuous vector fields on Rl, l ≥ 2, where their discontinuity set is a codimension one algebraic variety. By means of a regularizaron process proceeded by a blow-up technique we are able to bring about some results that bridge the space between discontinuous systems and singularly perturbed smooth systems. We also present an analysis of a subclass of discontinuous vector fields that present transient behavior in the 2-dimensional case, and we dedicate a section to providing sufficient conditions in order for our systems to have local asymptotic stability. © 2009 Society for Industrial and Applied Mathematics.
Original languageEnglish
Pages (from-to)508-526
JournalSIAM Journal on Applied Dynamical Systems
Volume8
Issue number1
DOIs
Publication statusPublished - 31 Mar 2009

Keywords

  • Discontinuous vector fields
  • Regularizaron
  • Singular perturbation
  • Sliding vector fields
  • Vector fields

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